On General Concavity Extensions of Grünbaum Type Inequalities

Given a strictly increasing continuous function \(\phi :\mathbb {R}_{\ge 0} \longrightarrow \mathbb {R}\cup \{-\infty \}\) with \(\lim _{t\rightarrow \infty }\phi (t) = \infty \), a function \(f:[a,b] \longrightarrow \mathbb {R}_{\ge 0}\) is said to be \(\phi \)-concave if \(\phi \circ f\) is concave. When \(\phi (t) = t^p\), \(p>0\), this notion is that of p-concavity whereas for \(\phi (t) = \log (t)\) it leads to the so-called log-concavity. In this work, we show that if the cross-sections volume function of a compact set \(K\subset \mathbb {R}^n\) (of positive volume) w.r.t. some hyperplane H passing through its centroid is \(\phi \)-concave, then one can find a sharp lower bound for the ratio \(\textrm{vol}(K^{-})/\textrm{vol}(K)\), where \(K^{-}\) denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. Moreover, other related results for \(\phi \)-concave functions (and involving the centroid) are shown.